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Mirrors > Home > MPE Home > Th. List > f1ocnvd | Structured version Visualization version GIF version |
Description: Describe an implicit one-to-one onto function. (Contributed by Mario Carneiro, 30-Apr-2015.) |
Ref | Expression |
---|---|
f1od.1 | ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶) |
f1od.2 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑊) |
f1od.3 | ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝐷 ∈ 𝑋) |
f1od.4 | ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ 𝑥 = 𝐷))) |
Ref | Expression |
---|---|
f1ocnvd | ⊢ (𝜑 → (𝐹:𝐴–1-1-onto→𝐵 ∧ ◡𝐹 = (𝑦 ∈ 𝐵 ↦ 𝐷))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1od.2 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑊) | |
2 | 1 | ralrimiva 3142 | . . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐶 ∈ 𝑊) |
3 | f1od.1 | . . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶) | |
4 | 3 | fnmpt 6639 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝑊 → 𝐹 Fn 𝐴) |
5 | 2, 4 | syl 17 | . . 3 ⊢ (𝜑 → 𝐹 Fn 𝐴) |
6 | f1od.3 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝐷 ∈ 𝑋) | |
7 | 6 | ralrimiva 3142 | . . . . 5 ⊢ (𝜑 → ∀𝑦 ∈ 𝐵 𝐷 ∈ 𝑋) |
8 | eqid 2736 | . . . . . 6 ⊢ (𝑦 ∈ 𝐵 ↦ 𝐷) = (𝑦 ∈ 𝐵 ↦ 𝐷) | |
9 | 8 | fnmpt 6639 | . . . . 5 ⊢ (∀𝑦 ∈ 𝐵 𝐷 ∈ 𝑋 → (𝑦 ∈ 𝐵 ↦ 𝐷) Fn 𝐵) |
10 | 7, 9 | syl 17 | . . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↦ 𝐷) Fn 𝐵) |
11 | f1od.4 | . . . . . . 7 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ 𝑥 = 𝐷))) | |
12 | 11 | opabbidv 5170 | . . . . . 6 ⊢ (𝜑 → {〈𝑦, 𝑥〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)} = {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐵 ∧ 𝑥 = 𝐷)}) |
13 | df-mpt 5188 | . . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)} | |
14 | 3, 13 | eqtri 2764 | . . . . . . . 8 ⊢ 𝐹 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)} |
15 | 14 | cnveqi 5829 | . . . . . . 7 ⊢ ◡𝐹 = ◡{〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)} |
16 | cnvopab 6090 | . . . . . . 7 ⊢ ◡{〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)} = {〈𝑦, 𝑥〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)} | |
17 | 15, 16 | eqtri 2764 | . . . . . 6 ⊢ ◡𝐹 = {〈𝑦, 𝑥〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)} |
18 | df-mpt 5188 | . . . . . 6 ⊢ (𝑦 ∈ 𝐵 ↦ 𝐷) = {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐵 ∧ 𝑥 = 𝐷)} | |
19 | 12, 17, 18 | 3eqtr4g 2801 | . . . . 5 ⊢ (𝜑 → ◡𝐹 = (𝑦 ∈ 𝐵 ↦ 𝐷)) |
20 | 19 | fneq1d 6593 | . . . 4 ⊢ (𝜑 → (◡𝐹 Fn 𝐵 ↔ (𝑦 ∈ 𝐵 ↦ 𝐷) Fn 𝐵)) |
21 | 10, 20 | mpbird 256 | . . 3 ⊢ (𝜑 → ◡𝐹 Fn 𝐵) |
22 | dff1o4 6790 | . . 3 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ ◡𝐹 Fn 𝐵)) | |
23 | 5, 21, 22 | sylanbrc 583 | . 2 ⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵) |
24 | 23, 19 | jca 512 | 1 ⊢ (𝜑 → (𝐹:𝐴–1-1-onto→𝐵 ∧ ◡𝐹 = (𝑦 ∈ 𝐵 ↦ 𝐷))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3063 {copab 5166 ↦ cmpt 5187 ◡ccnv 5631 Fn wfn 6489 –1-1-onto→wf1o 6493 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5255 ax-nul 5262 ax-pr 5383 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2888 df-ral 3064 df-rex 3073 df-rab 3407 df-v 3446 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-sn 4586 df-pr 4588 df-op 4592 df-br 5105 df-opab 5167 df-mpt 5188 df-id 5530 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 |
This theorem is referenced by: f1od 7602 f1ocnv2d 7603 pw2f1ocnv 41337 |
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